Why is moving objects, on the Z Plane, so unprecise?
When translating constrained to a plane, the mouse motion is translated relative to that plane. This is fine when the 'view' is relatively distant to the plane but if the view is close to the plane of motion then any small movement can be drastically amplified (for example, moving the mouse slightly 'up' can result in the object disappearing into the distance).

To test this, simply open the default scene. Select the cube and G to grab it, then Shift+Z. Note that when you move the mouse the motion of the cube closely matches. Cancel the movement by pressing Escape so the cube returns to the centre.

Now move the view so that you are close to the plane of the grid like so :



Now repeat the above keystrokes to grab the cube constrained to the plane. Move the mouse up/down and see how it now behaves - it will shoot off into the distance as the mouse moves up. The motion is relative to the plane - since the plane is at a very shallow angle in the view, small vertical motions correspond to large changes in distance.

To avoid this issue simply ensure the view point is not close to the plane of motion.